In which case, upon the end of term of the present administration, does the Speaker of the House automatically succeed to the Presidency instead?
No. The Vice-President (VP) would become President before the Speaker of the House. Note that the Senate chooses the VP before the House votes.
If the Senate can refuse to fill a seat on the Supreme Court, can the House refuse to fill the Presidency?
The Senate can't refuse to fill a seat on the Supreme Court. The Senate has no power to fill a seat on the Supreme Court. Their only power is to consent to the President filling such a seat. They can of course refuse such consent, otherwise there would be no need to ask them for consent.
And since the Speaker doesn't actually have to be an elected member of the House,
That's a possible interpretation of the law but has never been tested. All Speakers have been members of the House.
In theory, I guess that the House and Senate could manage this. However, it would be difficult for them to argue that the President picked in this way had been picked legitimately. People could reject presidential decisions on the basis of not having a legitimate president.
It's also not clear that the House has the right to move on to other business before choosing the President. And of course if there's no legitimate president, then there's no one to whom to send bills for signature. So Congress couldn't pass laws.